Math
Elimination Method Calculator
Type two or three linear equations and the elimination method calculator solves them the way it is taught: multiply the equations by numbers that make one variable's coefficients opposite, add to eliminate it, repeat until one variable remains, then back-substitute.
Every multiplied equation, every sum, and the final check are shown with exact fractions. Systems with no solution or infinitely many solutions are identified and explained.
Two equations in x and y, or three in x, y, and z. Each line needs one = sign. Fractions and decimals are fine.
Try an example
Result
Solution
—
- As an ordered tuple
- —
- Type of system
- —
- Check
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- Unknowns
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- Equations in standard form
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the elimination method calculator solves
The elimination (addition) method removes a variable by adding two equations whose The number multiplying a variable, like the 3 in 3x. for that variable are opposites. If they are not opposites already, multiply one or both equations by suitable numbers first. For three unknowns, eliminate the same variable from two pairs of equations, then eliminate a second variable from the resulting pair.
Steps for a 3×3 system
- Use equation (1) with (2), and (1) with (3), to eliminate x. That leaves two equations in y and z.
- Eliminate y from those two to get one equation in z, and solve it.
- Back-substitute z to find y, then y and z to find x.
- Check all three original equations.
| What elimination produces | Type | Solutions |
|---|---|---|
| a value for each variable | independent | exactly one |
| 0 = 0 (an equation vanishes) | dependent | infinitely many |
| 0 = k with k ≠ 0 | inconsistent | none |
How to use it
- Type the equations, one per line: two equations in x and y, or three in x, y, and z.
- Read the solution and the system type at the top.
- Open the steps to see each multiplied equation, each sum, the back-substitution, and the check.
How to read the answer
An independent system has exactly one solution, listed for each variable with its exact fraction. A dependent system shows the solution family with a free variable. An inconsistent system reports no solution and shows the false equation (such as 0 = 5) that revealed it.
Common mistakes and edge cases
- Multiplying only some terms of an equation. Every term, including the right side, must be multiplied.
- Adding equations whose chosen coefficients are equal instead of opposite. Multiply one by -1 first, or subtract.
- In 3×3 systems, eliminating different variables from the two pairs. Both pairs must drop the same variable.
- Reading 0 = 0 as a solution: it means the equation was redundant, not that a variable is 0.
Worked examples
2×2 with one multiplication
2x + 3y = 7 and 4x - y = 7 → x = 2, y = 1
Solution
x = 2, y = 1
2×2 that needs both equations multiplied
3x + 2y = 4 and 5x - 3y = 13 → x = 2, y = -1
Solution
x = 2, y = -1
3×3 system
x + y + z = 6, 2x - y + z = 3, x + 2y - z = 2 → (1, 2, 3)
Solution
x = 1, y = 2, z = 3
Fraction answer
2x + 5y = 1 and 3x - 4y = 2 → x = 14/23, y = -1/23
Solution
x = 14/23 (0.608696), y = -1/23 (-0.043478)
No solution
x + y = 2 and 2x + 2y = 5 → 0 = 1
Solution
No solution (inconsistent system)
Infinitely many solutions
x - y = 1 and 2x - 2y = 2 → 0 = 0
Solution
Infinitely many solutions: x = 1 + y (y free)
Wrong number of equations
Three unknowns need three equations
Solution
Error
Frequently asked questions
What is the elimination method?+
A way to solve linear systems by adding equations to cancel a variable. You first multiply the equations so that one variable has opposite coefficients, then add. It is also called the addition method or linear combination.
Which variable should I eliminate first?+
The one whose coefficients are easiest to make opposite: if they are already equal or opposite, or one is a multiple of the other. The calculator works through x first, then y, which always succeeds when the system is independent.
How does elimination work with three variables?+
Eliminate the same variable from two different pairs of equations to get a 2×2 system, solve that by elimination, then back-substitute to recover the third variable.
What if I get 0 = 0?+
That equation was a combination of the others, so the system is dependent and has infinitely many solutions. The calculator writes the solution family with a free variable.
What if I get 0 = 7?+
The equations contradict each other; there is no solution. For two variables the lines are parallel; for three, the planes have no common point.
Is elimination better than substitution?+
When equations are in standard form with no coefficient of 1, elimination avoids fractions and is faster. Substitution is easier when one equation is already solved for a variable.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026